Constructing a spreadsheet FFT power spectrum from daily prices

A daily series is prepared before the transform

The historical workflow used 95 daily December copper closes running from July 8, 1993 through November 18, 1993. Those ordered closes were not sent straight into a transform. They were first turned into a residual series, tapered with a Hanning window, and zero-padded so the fast Fourier transform could run on a power-of-two length.

Fourier analysis, in this construction, is the decomposition of the prepared observations into sinusoidal components that can then be plotted as a power spectrum. Spectral analysis is the inspection of that power across frequency or cycle length after the preprocessing and the transform. The fast Fourier transform is the discrete mapping into frequency bins once the input length is a power of two.

Remove the linear trend first

Prices were first converted to a residual series by subtracting a linear least-squares trend from each close. The detrend step is that removal of the linear least-squares fit from the price series before windowing.

Apply a Hanning window to the residuals

A Hanning window was built with a one-period increment of 2π divided by the sample count plus one, then applied as 0.5 times one minus the cosine of that increment times the observation index. That raised-cosine weight tapers the ends of the finite sample before the transform.

The window weights were multiplied by the detrended closes to produce the series actually passed to the transform.

Pad to a power-of-two length

The Fourier input and output ranges had to be a power of two, so the 95 windowed values were zero-padded to a 256-point array. Zero-padding means appending zeros so the transform length becomes an admissible power of two.

The same length rule was stated more generally. If the original sample count is below 64 the transform array should be 128 long. If the sample count is between 128 and 256 the array should be 512 long.

Read power on frequency and on cycle length

Each power-spectrum ordinate was the sum of the squared imaginary part and the squared real part of the corresponding complex Fourier bin. That squared magnitude is the vertical axis of the spectrum plot.

Frequency in cycles per year was stepped by 252 divided by 256, matching daily sampling, a 256-point transform, and 252 trading days in a year.

Cycle length was obtained by inverting frequency and multiplying by 252 so the same power values could be plotted against length instead of frequency. Cycle length is the reciprocal of frequency, scaled by the number of trading days in a year when the observations are daily.